A note on static spaces and related problems

Abstract

We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of the full divergence of the Cotton tensor for static metrics and metrics satisfying the critical point equation.

13 pages

References in corpus (1)

• Total Scalar Curvature and Harmonic Curvature

Cited by in corpus (26)

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• Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant
• Quantum reverse hypercontractivity: its tensorization and application to strong converses
• On the mass of static metrics with positive cosmological constant -- II
• On static three-manifolds with positive scalar curvature
• Bach-flat h-almost gradient Ricci solitons
• Vanishing conditions on Weyl tensor for Einstein-type manifolds
• Four dimensional static and related critical spaces with harmonic curvature
• Einstein-type structures, Besse's conjecture and a uniqueness result for a $\phi$-CPE metric in its conformal class
• On the volume functional of compact manifolds with boundary with harmonic Weyl tensor
• On the geometry of electrovacuum spaces in higher dimensions
• Electrostatic system with divergence-free Bach tensor and non-null cosmological constant
• On Static Manifolds and Related Critical Spaces with cyclic parallel Ricci tensor
• Static Manifolds with Boundary and Rigidity of Scalar Curvature and Mean Curvature
• Critical metrics of the volume functional on three-dimensional manifolds
• V-static spaces with positive isotropic curvature
• On Critical Point Equation of Compact Manifolds with Zero radial Weyl Curvature
• Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary
• Deformation of the Weighted Scalar Curvature
• Gap theorems on critical point equation of the total scalar curvature with divergence-free Bach tensor
• Brown-York mass and positive scalar curvature II - Besse's conjecture and related problems
• Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations
• Besse conjecture with vanishing conditions on the Weyl tensor
• Bach-flat noncompact steady quasi-Einstein manifolds
• Critical metrics of the total scalar curvature functional on 4-manifolds